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The Plot to Make Us Stupid

David Runciman on the Lottery

‘Why is it,’ asks the mathematician John Allen Paulos in his book about the pitfalls of innumeracy, ‘that a lottery ticket with the numbers 2 13 17 20 29 36 is for most people far preferable to one with the numbers 1 2 3 4 5 6?’ It is not an easy question to answer. All lotteries, after all, rely on a recognition by those who participate in them that the winning numbers are chosen at random, if only so that the participants can feel that their numbers have as good a chance of coming up as any others. People need to know it is random, because random translates as ‘fair’. However, all lotteries also rely on their participants having a sense that some sequences of numbers are more likely to come up than others. Once it is seen that all sequences have precisely the same chance of coming up as 1 2 3 4 5 6, the whole business of participation starts to look a lot less attractive. So participants fall back on, and are encouraged to fall back on, a belief that random-looking sequences are more likely to be chosen at random than sequences that look familiar. This belief is a delusion, but it is a peculiarly powerful one – even a probability theorist would have to be feeling fairly tough-minded to select, as an example of six numbers chosen at random, the second of Paulos’s sequences in preference to the first.

If the question Paulos asks is puzzling, what are we to make of the fact that, according to Camelot, organisers of Britain’s National Lottery, the most popular sequence of numbers chosen by its participants (or ‘players’, as their jargon would have it) is 1 2 3 4 5 6, which appears on over ten thousand tickets every week? This is truly baffling. It does not mean that 1 2 3 4 5 6 is preferred by most people to random-looking sequences – ten thousand is still only a tiny proportion of the total tickets sold. It does mean that 1 2 3 4 5 6 is much preferred to any given random-looking sequence, including the one given by Paulos. Are we to suppose that there is a small but committed band of hyper-realists out there, resolved to respect the laws of chance at all costs? If so, they are behaving in a very irrational manner. The National Lottery, as we all know, operates on a pool system, with the jackpot shared every week among all the winning ticket-holders. When 1 2 3 4 5 6 finally does come up (as it is likely to do sometime in the next 250,000 years or so), the winners will receive, at current values, between £1000 and £4000 each. The truth is that there can be no rational explanation for this sort of behaviour. If ten thousand players each week are choosing to spend their money in this way, it is simply further evidence of the mesmeric hold the Lottery has managed to exert. People are being mesmerised by numbers.

For example, what is the difference between these two randomly generated sequences, 7 17 23 32 38 42 and 12 15 26 44 46 49? The answer, perhaps surprisingly, is £17,280,000. The first sequence represents the numbers that dropped out of Camelot’s glorified tumble-drier (remember: random is fair) on the occasion of their second ever roll-over draw, on 15 January last year. The numbers were matched by 133 winners, each of whom received around £122,000. The second sequence is the one chosen for the third roll-over on 10 June. It was matched on only one ticket, the holder of which received £17.4 million. There is no statistical explanation for this discrepancy. On an average week, the jackpot should be shared between three or four tickets (there are around 14 million possible combinations of numbers and around 50 million tickets are sold). The reason for the huge number of winners on 15 January 1995, and the reason there have been many more weeks than was first expected with no winners at all, is that most players have a very clear idea of the sort of sequence that ought to win. It ought to be neat without looking too familiar, the little marks spread ‘randomly’ across the page – a couple of lowish numbers, a couple of medium numbers, a couple of highish numbers. This belief is no different from the belief of those who seem to feel that the winning sequence should look both neat and familiar (1 2 3 4 5 6). The intention is to increase the chance of winning; the result is that the chances of winning the entire jackpot are greatly reduced. The only rational strategy to adopt when playing the Lottery is to wait for a roll-over (i.e. until the jackpot gets above £14 million), then choose a sequence that is unlikely to be chosen by anyone else (say, 32 33 35 36 37 39). However, the fact that some sequences are more likely to be chosen than others suggests that ‘choice’ is perhaps too strong a word for what most people are doing. In effect, their numbers are choosing themselves.

The full text of this essay is only available to subscribers of the London Review of Books.

Letters

In his antic and meticulously-observed spoof of a puritan coming up to barking point over the National Lottery, David Runciman says that the popularity of the combination 1 2 3 4 5 6 is ‘truly baffling’ (LRB, 22 February). Since the Lottery started I have bought three tickets. The first two times I went for 1 2 3 4 5 6 as I thought that this was the combination least likely to be chosen by others, and that it would maximise my expected return by minimising the chance of my having to share any winnings. I then discovered that many other people reason as I had, and that 1 2 3 4 5 6 is consequently the most popular choice. I didn’t use it for my third ticket. David Runciman makes the same point when he says that one should ‘choose a sequence that is unlikely to be chosen by anyone else’. A Brazilian friend once told me that there is a non-random South American lottery based entirely on this principle. The winning tickets each week are the ones bearing that week’s least popular choices of numbers. This introduces a subtle, entertaining and competitive element of second-guessing into the game that perhaps Camelot should bear in mind. In the meantime I have adopted a strategy that guarantees a lottery win every week: I have decided to buy some shares in Camelot’s holding companies.